I got stuck on the following question whilst learning about basic option pricing.
A stock is valued at \$75 today. An option will pay \$1 the first time the stock reaches \$100 in value, which it is assumed will happen with probability one at some point in the future. Find the price of the option and construct a replicating portfolio.
At first glance, this seems like some sort of continuous time problem, but I'm hoping that there's a simpler way to do things. How should one approach this sort of question?
Edit: For simplicity, let's assume there's no interest in this scenario.
Answer
Assume the stock pays no dividends before 100 dollars is hit. Interest rates can be arbitrary. Buy 1/100 of a share for 75 cents. Hold until $100 is hit then sell. The payoff of 1 dollar is replicated for an upfront cost of 75 cents. The arbitrage-free value of the option is 75 cents.
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